Team:UT-Tokyo/Data/Modeling/Model02

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{{:Team:UT-Tokyo/Templates/BeginContent|fullpagename=Team:UT-Tokyo/Data/Modeling/Model02|subpagename=Model2: A simple model for E.coli chemotaxis}}
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{{:Team:UT-Tokyo/Templates/BeginContent|fullpagename=Team:UT-Tokyo/Data/Modeling/Model02|subpagename=Model02}}
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=[[Team:UT-Tokyo/Data/Modeling|Modeling]]/Model2: A simple model for ''E.coli'' chemotaxis=
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=Aim=
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=[[Team:UT-Tokyo/Data/Modeling/Model02/applet|Interactive demo]]=
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We derived a simple relationship between Asp concentration and E.coli chemotaxis so that
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we were able to simulate macroscopic ¥(n~10^8) E.coli colony chemotacitc behavior.
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=Background=
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==Aim==
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We considered internal condition of E.coli (receptor methylation etc) by SPECS model<html><sup class="ref">[1]</sup></html>.
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We tried to derive a simple relationship between Asp concentration and ''E.coli'' chemotactic behavior so that
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SPECS is a relatively accurate taxis model, and is suitable for simulation of approximately ten thousand E.coli for approximately 10 minutes.
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it become possible to simulate macroscopic (n~10<html><sup>8</sup></html>) ''E.coli''-colony chemotactic behavior in the entire simulation.
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This model, however, take too much time to reproduce our system, because the system includes much more E.coli(about one hundred of million) and takes a relatively long time (a few days).
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So, by adapting SPECS model to our system, we have devised a new model that provides less calculation amount and enough accuracy.
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Concretely, we divided chemotactic motion of E.coli groups into two factors, that is, parallel translation and diffusion. Then we derived approximation equations that figure out the two respective values from Asp concentration and its gradient.
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=Method=
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==Background==
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In this study, we used SPECS method to simulate E.coli chemotactic motion.
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SPECS <html><sup class="ref">[1]</sup></html> is one of the models simulating ''E.coli'' chemotactic behavior, and it considers internal condition of ''E.coli'' (receptor methylation etc).  It is suitable for semi-macroscopic simulation which has around ten thousand ''E.coli'' and the timescale is within a hour.
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Here we introduce the brief overview of SPECS.
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For our system, however, it takes too much time to replicate entire system, because the system includes much more ''E.coli''(about one hundred of million) and the time scale is relatively long (a few days).
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Therefore we devised a new model based on SPECS which requires less calculation amount.
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We processed Monte Carlo simulation for 20000 E.colis which behavior was based on SPECS in some different Asp concentration and its gradient.  The gradient we used were exponential ¥[eqn 1].
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==Method==
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We approximated E.colis movement as the combination of parallel translation and diffusion.
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We approximated the motion of ''E.coli'' groups doing a process of chemotaxis into two factors, that is, parallel translation and diffusion.
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We recorded the time course of average position <x> and root-mean-square <<html>x<sup>2</sup></html>> and then
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calculated average velocity ¥[eqn 2] and the diffusion constant ¥[eqn 3].
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For these results we derived the relationship between:
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[[File:utt_m2_figure.PNG]]
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* Asp concentration and average velocity of E.coli
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* Asp concentration and diffusion constant of E.coli
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* gradient of Asp concentration and average velocity of E.coli
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* gradient of Asp concentration and diffusion constant of E.coli
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Finally we fitted approximated curve to these data.
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Let 'V' represent the parallel translation velocity, 'D' represent the diffusion coefficient of ''E.coli'' groups and 'A' represent L-Asp concentration.
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''E.coli'' can detect A and the gradient ∇A.
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They have an internal circuit to translate these signals into motions of their flagella and try to move towards regions with more Asp.
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So we thought V and D could be represented as a function of A and ∇A.
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And then we used the SPECS model to derive these functions.
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=Result=
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Here we introduce SPECS briefly.
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¥[fig1. relationship between Asp conc. and avg. velocity]
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An ''E.coli'' belongs to one of two mobile states: "Run" and "Tumble".
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¥[fig2. relationship between gradient of Asp conc. and avg. velocity]
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Run state indicates that ''E.coli'' goes straight until the state changes.
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¥[fig3. relationship between Asp conc. and diffusion coef.]
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Tumble state indicates that ''E.coli'' is changing its direction and doesn't move around.
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¥[fig4. relationship between gradient of Asp conc. and diffusion coef.]
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''E.coli'' changes its state from "Run" to "Tumble" with probability P<html><sub>rt</sub> </html> [1/sec].
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fitted curveは次の式で与えられる。
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From "Tumble" to "Run" with P<html><sub>tr</sub></html> [1/sec].
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¥[eqn4](tanh(0.66*(log(A)+11.35))+1.0)*0.055*grad(log(A))
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These probabilities depend on the ''E.coli'' internal state.
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¥[eqn5](tanh(0.65*(log(A)+11.5))+1.0)*(tanh(0.71*(grad(log(A))-2.05)+0.9)*0.0037+0.0047
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The internal state is represented by two variables: "a" and "m".
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¥φ means Asp concentration.
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"a" indicates kinase activity and "m" represents the receptor methylation level of the cell.
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These vary depending on L-Asp concentration.
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In addition, SPECS takes into account Brownian fluctuation.
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Using this model, SPECS tries to replicate the internal circuit of ''E.coli'' which translates an Asp concentration distribution into a mobile state.
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We processed a Monte Carlo simulation for 20000 ''E.coli'' behaving according to SPECS when placed in fields containing different concentrations and gradients of Asp.  The shape of the Asp gradient used was exponential according to the following relationship:
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[[File:utt_m2_eqn1.png]]
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where A0 represents the concentration of Asp at x = 0 and x0 a parameter that represents the slope of the gradient.
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We approximated ''E.coli'' movement by a combination of parallel translation and diffusion.
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We recorded the average position <x> and root-mean-square <x<html><sup>2</sup></html>> over time and then
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calculated the average velocity:
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[[File:utt_m2_eqn2.png]]
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and the diffusion constant:
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[[File:utt_m2_eqn3.png]]
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We derived the relationship between
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* Asp concentration and average velocity of ''E.coli''
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* Asp concentration and diffusion constant of ''E.coli''
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* gradient of Asp concentration and average velocity of ''E.coli''
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* gradient of Asp concentration and diffusion constant of ''E.coli''
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Finally we fitted approximated curves (represented as functions of A and ∇A) to these data so that we could derive functions we needed in the entire system simulation.
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==Result==
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{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig1.png|caption=Figure 1. relationship between Asp conc. and avg. velocity}}
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{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig2.png|caption=Figure 2. relationship between gradient of Asp conc. and avg. velocity}}
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{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig3.png|caption=Figure 3. relationship between Asp conc. and diffusion coef}}
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{{:Team:UT-Tokyo/Templates/Image|file=utt_m2_fig4.png|caption=Figure 4. relationship between gradient of Asp conc. and diffusion coef}}
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The fitted curves are represented by the following equations.
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[[File:utt_m2_eqn4.png]]
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[[File:utt_m2_eqn5.png]]
<html>
<html>
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<div id="references">
<div id="references">
<ul>
<ul>
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   <li id="ref_1">[1] Jiang L, Ouyang Q, Tu Y (2010) Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time. PLoS Comput Biol 6(4): e1000735. doi:10.1371/journal.pcbi.1000735</li>
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   <li id="ref_1">[1] Jiang L, Ouyang Q, Tu Y (2010) Quantitative Modeling of "Escherichia coli" Chemotactic Motion in Environments Varying in Space and Time. PLoS Comput Biol 6(4): e1000735. doi:10.1371/journal.pcbi.1000735</li>
</ul>
</ul>
</div>
</div>
</html>
</html>
{{:Team:UT-Tokyo/Templates/EndContent}}
{{:Team:UT-Tokyo/Templates/EndContent}}

Latest revision as of 02:08, 6 October 2011

Modeling/Model2: A simple model for E.coli chemotaxis

Interactive demo

Aim

We tried to derive a simple relationship between Asp concentration and E.coli chemotactic behavior so that it become possible to simulate macroscopic (n~108) E.coli-colony chemotactic behavior in the entire simulation.

Background

SPECS [1] is one of the models simulating E.coli chemotactic behavior, and it considers internal condition of E.coli (receptor methylation etc). It is suitable for semi-macroscopic simulation which has around ten thousand E.coli and the timescale is within a hour. For our system, however, it takes too much time to replicate entire system, because the system includes much more E.coli(about one hundred of million) and the time scale is relatively long (a few days). Therefore we devised a new model based on SPECS which requires less calculation amount.

Method

We approximated the motion of E.coli groups doing a process of chemotaxis into two factors, that is, parallel translation and diffusion.

Utt m2 figure.PNG

Let 'V' represent the parallel translation velocity, 'D' represent the diffusion coefficient of E.coli groups and 'A' represent L-Asp concentration. E.coli can detect A and the gradient ∇A. They have an internal circuit to translate these signals into motions of their flagella and try to move towards regions with more Asp. So we thought V and D could be represented as a function of A and ∇A. And then we used the SPECS model to derive these functions.

Here we introduce SPECS briefly. An E.coli belongs to one of two mobile states: "Run" and "Tumble". Run state indicates that E.coli goes straight until the state changes. Tumble state indicates that E.coli is changing its direction and doesn't move around. E.coli changes its state from "Run" to "Tumble" with probability Prt [1/sec]. From "Tumble" to "Run" with Ptr [1/sec]. These probabilities depend on the E.coli internal state. The internal state is represented by two variables: "a" and "m". "a" indicates kinase activity and "m" represents the receptor methylation level of the cell. These vary depending on L-Asp concentration. In addition, SPECS takes into account Brownian fluctuation. Using this model, SPECS tries to replicate the internal circuit of E.coli which translates an Asp concentration distribution into a mobile state.

We processed a Monte Carlo simulation for 20000 E.coli behaving according to SPECS when placed in fields containing different concentrations and gradients of Asp. The shape of the Asp gradient used was exponential according to the following relationship:

Utt m2 eqn1.png

where A0 represents the concentration of Asp at x = 0 and x0 a parameter that represents the slope of the gradient.

We approximated E.coli movement by a combination of parallel translation and diffusion. We recorded the average position <x> and root-mean-square <x2> over time and then calculated the average velocity:

Utt m2 eqn2.png

and the diffusion constant:

Utt m2 eqn3.png

We derived the relationship between

  • Asp concentration and average velocity of E.coli
  • Asp concentration and diffusion constant of E.coli
  • gradient of Asp concentration and average velocity of E.coli
  • gradient of Asp concentration and diffusion constant of E.coli

Finally we fitted approximated curves (represented as functions of A and ∇A) to these data so that we could derive functions we needed in the entire system simulation.

Result

Figure 1. relationship between Asp conc. and avg. velocity
Figure 1. relationship between Asp conc. and avg. velocity

Figure 2. relationship between gradient of Asp conc. and avg. velocity
Figure 2. relationship between gradient of Asp conc. and avg. velocity

Figure 3. relationship between Asp conc. and diffusion coef
Figure 3. relationship between Asp conc. and diffusion coef

Figure 4. relationship between gradient of Asp conc. and diffusion coef
Figure 4. relationship between gradient of Asp conc. and diffusion coef

The fitted curves are represented by the following equations.

Utt m2 eqn4.png

Utt m2 eqn5.png

References

  • [1] Jiang L, Ouyang Q, Tu Y (2010) Quantitative Modeling of "Escherichia coli" Chemotactic Motion in Environments Varying in Space and Time. PLoS Comput Biol 6(4): e1000735. doi:10.1371/journal.pcbi.1000735

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